Sub-Claim 1: Multiplication modulo 24 is associative on the set R.
Proof: Multiplication is associative on the integers and every element of the set R is an integer so associativity holds. The fact that our operation is multiplication modulo 24, rather than normal multiplication, does not change this because, in an equation with modular multiplication, only the final product changes, and two equal products are changed to the same value.

Therefore, U(24) is a group because it fulfills all three of the group axioms.

Because every element of R is its own inverse, any subset of R, which is not all of R and which includes 1 and at least one other element is a non-trivial, proper subgroup. I hope you don't want me to list them all out because there are a lot; 2^7 if I remember my Combinatorics correctly. A few examples though are: